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\begin{array}{c}\phantom{\times999}999999\\\underline{\times\phantom{999999}876}\\\end{array}
First line up the numbers vertically and match the places from the right like this.
\begin{array}{c}\phantom{\times999}999999\\\underline{\times\phantom{999999}876}\\\phantom{\times99}5999994\\\end{array}
Now multiply the first number with the 1^{st} digit in 2^{nd} number to get intermediate results. That is Multiply 999999 with 6. Write the result 5999994 at the end leaving 0 spaces to the right like this.
\begin{array}{c}\phantom{\times999}999999\\\underline{\times\phantom{999999}876}\\\phantom{\times99}5999994\\\phantom{\times9}6999993\phantom{9}\\\end{array}
Now multiply the first number with the 2^{nd} digit in 2^{nd} number to get intermediate results. That is Multiply 999999 with 7. Write the result 6999993 at the end leaving 1 spaces to the right like this.
\begin{array}{c}\phantom{\times999}999999\\\underline{\times\phantom{999999}876}\\\phantom{\times99}5999994\\\phantom{\times9}6999993\phantom{9}\\\underline{\phantom{\times}7999992\phantom{99}}\\\end{array}
Now multiply the first number with the 3^{rd} digit in 2^{nd} number to get intermediate results. That is Multiply 999999 with 8. Write the result 7999992 at the end leaving 2 spaces to the right like this.
\begin{array}{c}\phantom{\times999}999999\\\underline{\times\phantom{999999}876}\\\phantom{\times99}5999994\\\phantom{\times9}6999993\phantom{9}\\\underline{\phantom{\times}7999992\phantom{99}}\\\phantom{\times}875999124\end{array}
Now add the intermediate results to get final answer.